A new approach to interpolation of compact linear operators

TL;DR

This paper presents an abstract theorem ensuring the preservation of compactness of linear operators during interpolation in Banach spaces, using a novel approach that simplifies proofs and broadens applicability.

Contribution

It introduces a new abstract framework for interpolation of compact operators, avoiding complex analytical details and allowing the complex method as a special case.

Findings

Proves the preservation of compactness under interpolation in Banach spaces.

Uses a reductio ad absurdum approach with specially constructed subspaces.

Applicable to the complex interpolation method as a particular case.

Abstract

We prove an abstract theorem on keeping the compactness property of a linear operator after interpolation in Banach spaces. Our approach consists of two features. Applying the principle "reductio ad absurdum," we obtain a possibility to carry out all proofs only for some specially constructed subspaces of the given spaces, e.g., having a common Schauder basis. As a second feature, we consider in all assertions only embedding operators obtaining the full result just at the end of the paper. No analytical presentation of operators, spaces and interpolation functors is required and the complex method is admissible as a particular case.